Group #15675089 — subgroups

Subgroups — #15675089

Order 2944.

262 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092401	C₂	1	2	yes	maximal, complemented
2	#13092214	C₂	1	4	yes	maximal, complemented
3	C₂₃×#535	C₂	1	1	yes	maximal, characteristic
4	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂₃×#127	C₄	1	2	yes	—
7	C₂₃×#124	C₂²	1	2	yes	—
8	C₂₃×#121	—	2	2	no	—
9	C₂₃×#120	—	2	2	no	—
10	C₂₃×#117	C₄	1	1	yes	characteristic
11	C₂×#67267	—	2	2	no	—
12	#10625018	C₂²	1	4	yes	complemented
13	#10625015	—	2	2	no	—
14	C₂⁴×C₂₃	—	2	1	no	—
15	#41×C₂₃	—	4	2	no	—
16	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
17	C₂×D₄×C₂₃	—	2	4	no	—
18	C₂×D₄×C₂₃	—	2	1	no	—
19	C₂×D₄×C₂₃	—	4	2	no	—
20	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂²⋊C₄×C₂₃	—	2	1	no	—
27	C₂²⋊C₄×C₂₃	—	2	4	no	—
28	C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	#67267	—	2	4	no	—
30	#67267	—	4	2	no	—
31	C₂×C₂₃⋊C₈	—	4	2	no	—
32	C₂³×C₂₃	—	2	2	no	—
33	C₂³×C₂₃	—	4	2	no	—
34	C₂³×C₂₃	—	8	2	no	—
35	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
37	Q₈×C₂₃	—	4	1	no	—
38	D₄×C₂₃	—	4	1	no	—
39	D₄×C₂₃	—	4	2	no	—
40	D₄×C₂₃	—	4	2	no	—
41	D₄×C₂₃	—	4	2	no	—
42	D₄×C₂₃	—	8	4	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂₃⋊C₈	—	4	4	no	—
51	#2134	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	—	4	2	no	—
55	C₂²×C₂₃	—	4	1	no	—
56	C₂²×C₂₃	—	4	4	no	—
57	C₂²×C₂₃	—	4	2	no	—
58	C₂²×C₂₃	—	8	4	no	—
59	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	4	2	no	—
63	C₄×C₂₃	#99	1	2	yes	—
64	#535	D₂₃	1	1	yes	characteristic
65	#413	—	23	2	no	—
66	#355	—	23	4	no	—
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	—	4	2	no	—
69	C₂×C₂₃	—	4	1	no	—
70	C₂×C₂₃	—	8	2	no	—
71	C₂×C₂₃	C₂×#99	1	1	yes	characteristic
72	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
73	C₂²×D₄	Dic₂₃	1	1	yes	characteristic
74	C₂×M₁₆	—	46	2	no	—
75	#127	Dic₂₃	1	2	yes	—
76	#124	C₂×D₂₃	1	2	yes	—
77	#121	—	2	2	no	—
78	#120	—	2	2	no	—
79	#117	Dic₂₃	1	1	yes	characteristic
80	#101	—	23	4	no	—
81	#100	—	46	2	no	—
82	C₂₃	#2134	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	—	2	1	no	—
84	#41	—	4	2	no	—
85	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
86	C₂×D₄	—	2	4	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
90	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
91	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
92	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
94	M₁₆	—	46	4	no	—
95	M₁₆	—	92	2	no	—
96	C₂×C₈	—	92	2	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₂²⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	C₂×Dic₂₃	1	2	yes	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	8	2	no	—
104	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
105	C₂³	C₂²⋊Dic₂₃	1	2	yes	—
106	Q₈	—	4	1	no	—
107	D₄	—	4	1	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
119	C₈	—	92	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	4	2	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	8	4	no	—
127	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	4	2	no	—
131	C₄	#10625016	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	4	1	no	—
134	C₂	—	4	2	no	—
135	C₂	—	8	2	no	—
136	C₂	C₂×#10625016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092401

C₂

yes

maximal, complemented

#13092214

C₂

yes

maximal, complemented

C₂₃×#535

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₄

yes

characteristic

C₂₃×#127

C₄

yes

—

C₂₃×#124

C₂²

yes

—

C₂₃×#121

—

C₂₃×#120

—

C₂₃×#117

C₄

yes

characteristic

C₂×#67267

—

#10625018

C₂²

yes

complemented

#10625015

—

C₂⁴×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

characteristic

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

C₂³×C₂₃

C₂²⋊C₄

yes

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#2134

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#99

yes

—

#535

D₂₃

yes

characteristic

#413

—

#355

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#99

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

C₂²×D₄

Dic₂₃

yes

characteristic

C₂×M₁₆

—

#127

Dic₂₃

yes

—

#124

C₂×D₂₃

yes

—

#121

—

#120

—

#117

Dic₂₃

yes

characteristic

#101

—

#100

—

C₂₃

#2134

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

C₂²×D₂₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂×D₄

C₂×Dic₂₃

yes

characteristic

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

C₂×Dic₂₃

yes

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂²⋊Dic₂₃

yes

characteristic

105

C₂³

C₂²⋊Dic₂₃

yes

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

117

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#10625016

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

C₂×#10625016

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#10625016
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₂₃
Fitting, prime radical	C₂₃×#535	C₂
socle	C₂×C₂₃	C₂×#99
cosocle	C₂×Q₈×C₂₃	C₂³
hypercenter	#535	D₂₃
hypocenter	C₂₃	#2134
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #535

subgroup

G/H

center

C₂

C₂×#10625016

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₂₃

Fitting, prime radical

C₂₃×#535

C₂

socle

C₂×C₂₃

C₂×#99

cosocle

C₂×Q₈×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #535

N	H ≅ G/N	type
1. #13092401	C₂	semidirect
2. #13092214	C₂	semidirect
12. #10625018	C₂²	semidirect
82. C₂₃	#2134	semidirect

H ≅ G/N

type

1. #13092401

C₂

semidirect

2. #13092214

C₂

semidirect

12. #10625018

C₂²

semidirect

82. C₂₃

#2134

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092401	2
#13092214	2
C₂₃×#535	2
#2134	23

subgroup

index

#13092401

#13092214

C₂₃×#535

#2134

Subgroups — #15675089

Conjugacy classes of subgroups

Complements

Maximal subgroups